Group #46819952 — subgroups

Subgroups — #46819952

Order 34432.

410 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 205 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770683	C₂	1	4	yes	maximal, complemented
2	D₄×#14524130	C₂	1	4	yes	maximal, complemented
3	#31770658	C₂	1	2	yes	maximal, complemented
4	#31770572	C₂	1	4	yes	maximal, complemented
5	#532×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864386	C₂²	1	4	yes	—
7	C₂²×#14524130	C₂²	1	4	yes	—
8	C₄⋊#14524130	C₂²	1	2	yes	—
9	C₄⋊#14524130	C₂²	1	2	yes	—
10	C₂²⋊#14524130	C₂²	1	4	yes	—
11	C₂²⋊#14524130	C₂²	1	8	yes	—
12	C₄×#14524130	C₂²	1	4	yes	—
13	C₂×#41×C₂₆₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₂₆₉	C₄	1	1	yes	characteristic
15	#127×C₂₆₉	C₄	1	1	yes	characteristic
16	#124×C₂₆₉	C₄	1	2	yes	—
17	#121×C₂₆₉	C₄	1	4	yes	—
18	C₄×D₄×C₂₆₉	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
20	#16864386	—	4	4	no	—
21	C₂×#14524130	—	4	4	no	—
22	C₂×#14524130	C₂³	1	4	yes	—
23	C₂×#14524130	C₂³	1	4	yes	—
24	#41×C₂₆₉	—	4	2	no	—
25	#41×C₂₆₉	—	4	2	no	—
26	C₂×Q₈×C₂₆₉	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	—
28	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₂₆₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
33	C₄⋊C₄×C₂₆₉	C₂×C₄	1	4	yes	—
34	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	—
35	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₂₆₉	C₂³	1	2	yes	—
38	#14524130	—	2	4	no	—
39	#14524130	—	2	4	no	—
40	C₂³×C₂₆₉	C₂²×C₄	1	4	yes	—
41	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₂₆₉	—	4	2	no	—
43	D₄×C₂₆₉	—	2	8	no	—
44	D₄×C₂₆₉	—	4	2	no	—
45	D₄×C₂₆₉	—	4	4	no	—
46	C₂×C₄×C₂₆₉	—	2	4	no	—
47	C₂×C₄×C₂₆₉	—	2	1	no	—
48	C₂×C₄×C₂₆₉	—	2	1	no	—
49	C₂×C₄×C₂₆₉	—	2	4	no	—
50	C₂×C₄×C₂₆₉	—	2	4	no	—
51	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
56	C₂²×C₂₆₉	—	2	1	no	—
57	C₂²×C₂₆₉	—	2	4	no	—
58	C₂²×C₂₆₉	—	2	2	no	—
59	C₂²×C₂₆₉	—	2	8	no	—
60	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₂₆₉	—	2	1	no	—
62	C₄×C₂₆₉	—	2	4	no	—
63	C₄×C₂₆₉	—	2	2	no	—
64	C₄×C₂₆₉	#131	1	4	yes	—
65	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂×C₂₆₉	—	4	1	no	—
68	C₂×C₂₆₉	—	4	4	no	—
69	C₂×C₂₆₉	#518	1	1	yes	characteristic
70	C₂×C₂₆₉	C₂×#131	1	2	yes	—
71	C₂₆₉	#2979	1	1	yes	characteristic, complemented, Sylow
72	#2979	—	269	1	no	maximal, Sylow
73	#532	D₂₆₉	1	1	yes	characteristic
74	#436	—	269	4	no	—
75	C₈×D₄	—	269	4	no	—
76	#431	—	269	2	no	—
77	#408	—	269	4	no	—
78	C₂×#41	C₂×D₂₆₉	1	2	yes	—
79	C₂×M₁₆	—	269	4	no	—
80	C₂²×C₈	—	269	4	no	—
81	C₄⋊Q₈	Dic₂₆₉	1	1	yes	characteristic
82	#127	Dic₂₆₉	1	1	yes	characteristic
83	#124	Dic₂₆₉	1	2	yes	—
84	#121	Dic₂₆₉	1	4	yes	—
85	C₄×D₄	C₂×D₂₆₉	1	4	yes	—
86	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
87	C₄⋊C₈	—	269	2	no	—
88	C₄⋊C₈	—	269	2	no	—
89	C₂²⋊C₈	—	269	4	no	—
90	C₂²⋊C₈	—	269	8	no	—
91	C₄×C₈	—	269	4	no	—
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	C₂×Dic₂₆₉	1	2	yes	—
95	C₂×D₄	C₂×Dic₂₆₉	1	4	yes	—
96	C₂×D₄	C₂×Dic₂₆₉	1	4	yes	—
97	C₂×D₄	C₂×Dic₂₆₉	1	2	yes	—
98	C₂²×C₄	C₂×Dic₂₆₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
101	M₁₆	—	1076	4	no	—
102	C₂×C₈	—	269	4	no	—
103	C₂×C₈	—	269	4	no	—
104	C₂×C₈	—	1076	4	no	—
105	C₄⋊C₄	C₂×Dic₂₆₉	1	4	yes	—
106	C₂²⋊C₄	C₂×Dic₂₆₉	1	8	yes	—
107	C₄²	C₂×Dic₂₆₉	1	1	yes	characteristic
108	C₄²	C₂×Dic₂₆₉	1	1	yes	characteristic
109	C₄²	C₂²×D₂₆₉	1	2	yes	—
110	C₂³	C₂²×Dic₂₆₉	1	4	yes	—
111	C₂³	C₂²×Dic₂₆₉	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
123	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
124	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
125	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
126	C₈	—	538	4	no	—
127	C₈	—	538	4	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	#22575460	1	4	yes	—
137	C₄	#22575498	1	1	yes	characteristic
138	C₄	#22575498	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	4	4	no	—
141	C₂	#31771080	1	1	yes	characteristic
142	C₂	C₂×#22575460	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31770683

C₂

yes

maximal, complemented

D₄×#14524130

C₂

yes

maximal, complemented

#31770658

C₂

yes

maximal, complemented

#31770572

C₂

yes

maximal, complemented

#532×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

C₂²

yes

—

C₂²×#14524130

C₂²

yes

—

C₄⋊#14524130

C₂²

yes

—

C₄⋊#14524130

C₂²

yes

—

C₂²⋊#14524130

C₂²

yes

—

C₂²⋊#14524130

C₂²

yes

—

C₄×#14524130

C₂²

yes

—

C₂×#41×C₂₆₉

C₂²

yes

—

C₄⋊Q₈×C₂₆₉

C₄

yes

characteristic

#127×C₂₆₉

C₄

yes

characteristic

#124×C₂₆₉

C₄

yes

—

#121×C₂₆₉

C₄

yes

—

C₄×D₄×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

—

—

C₂³

yes

—

C₂×#14524130

C₂³

yes

—

#41×C₂₆₉

—

#41×C₂₆₉

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

—

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

C₂×C₄

yes

—

C₂²⋊C₄×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

C₂×C₄

yes

characteristic

C₄²×C₂₆₉

C₂×C₄

yes

characteristic

C₄²×C₂₆₉

C₂³

yes

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

C₂²×C₄

yes

—

C₂³×C₂₆₉

C₂²×C₄

yes

characteristic

Q₈×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#131

yes

—

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#518

yes

characteristic

C₂×C₂₆₉

C₂×#131

yes

—

C₂₆₉

#2979

yes

characteristic, complemented, Sylow

#2979

—

269

maximal, Sylow

#532

D₂₆₉

yes

characteristic

#436

—

269

—

C₈×D₄

—

269

—

#431

—

269

—

#408

—

269

—

C₂×#41

C₂×D₂₆₉

yes

—

C₂×M₁₆

—

269

—

C₂²×C₈

—

269

—

C₄⋊Q₈

Dic₂₆₉

yes

characteristic

#127

Dic₂₆₉

yes

characteristic

#124

Dic₂₆₉

yes

—

#121

Dic₂₆₉

yes

—

C₄×D₄

C₂×D₂₆₉

yes

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

C₄⋊C₈

—

269

—

C₄⋊C₈

—

269

—

C₂²⋊C₈

—

269

—

C₂²⋊C₈

—

269

—

C₄×C₈

—

269

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₂₆₉

yes

—

C₂×D₄

C₂×Dic₂₆₉

yes

—

C₂×D₄

C₂×Dic₂₆₉

yes

—

C₂×D₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

C₂×Dic₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

100

C₂²×C₄

C₂²×D₂₆₉

yes

—

101

M₁₆

—

1076

—

102

C₂×C₈

—

269

—

103

C₂×C₈

—

269

—

104

C₂×C₈

—

1076

—

105

C₄⋊C₄

C₂×Dic₂₆₉

yes

—

106

C₂²⋊C₄

C₂×Dic₂₆₉

yes

—

107

C₄²

C₂×Dic₂₆₉

yes

characteristic

108

C₄²

C₂×Dic₂₆₉

yes

characteristic

109

C₄²

C₂²×D₂₆₉

yes

—

110

C₂³

C₂²×Dic₂₆₉

yes

—

111

C₂³

C₂²×Dic₂₆₉

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

122

C₂×C₄

C₂²×Dic₂₆₉

yes

—

123

C₂×C₄

C₂²×Dic₂₆₉

yes

—

124

C₂×C₄

C₂²×Dic₂₆₉

yes

—

125

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

126

C₈

—

538

—

127

C₈

—

538

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₂₆₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#22575460

yes

—

137

C₄

#22575498

yes

characteristic

138

C₄

#22575498

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#31771080

yes

characteristic

142

C₂

C₂×#22575460

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting, prime radical	#532×C₂₆₉	C₂
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#532	D₂₆₉
hypocenter	C₂₆₉	#2979
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₆₉

commutator, socle

C₂²×C₂₆₉

C₂³×C₄

Fitting, prime radical

#532×C₂₆₉

C₂

cosocle

C₂×C₄×C₂₆₉

C₂⁴

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #31770683	C₂	semidirect
2. D₄×#14524130	C₂	semidirect
3. #31770658	C₂	semidirect
4. #31770572	C₂	semidirect
71. C₂₆₉	#2979	semidirect

H ≅ G/N

type

1. #31770683

C₂

semidirect

2. D₄×#14524130

C₂

semidirect

3. #31770658

C₂

semidirect

4. #31770572

C₂

semidirect

71. C₂₆₉

#2979

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#31770683	2
D₄×#14524130	2
#31770658	2
#31770572	2
#532×C₂₆₉	2
#2979	269

subgroup

index

#532×C₂₆₉

269

Subgroups — #46819952

Conjugacy classes of subgroups

Complements

Maximal subgroups